Medieval Islamic tiling reveals mathematical savvy

Girih strapwork pattern on an interior archway in the Sultan’s Lodge in the Green Mosque in Bursa, Turkey (AD 1424)

(Image: W B Denny)

Girih-tile reconstruction of the girih strapwork pattern on an interior archway in the Sultan’s Lodge in the Green Mosque in Bursa, Turkey (AD 1424)

(Image: W B Denny)

Medieval Islamic designers used elaborate geometrical tiling patterns at least 500 years before Western mathematicians developed the concept.

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The geometric design, called “girih”, was widely used to decorate Islamic buildings but the advanced mathematical concept within the patterns was not recognised, until now. Physicist Peter Lu at Harvard University in Cambridge, Massachusetts, US, realised the 15th-century tiles formed so-called Penrose geometric patterns, when he spotted them on a visit to Uzbekistan.

Scholars had thought the girih were created by drawing a zigzag network of lines with a straight edge and compass. But when Lu looked at them, he recognised the regular but non-repetitive patterns of Penrose tiling – a concept developed in the West only in the 1970s.

Simple periodic patterns can be generated easily by repeating a unit cell of several elements, a technique widely used in tile patterns, but the rotational symmetry possible is limited. In the 1970s, Roger Penrose at the University of Oxford in the UK showed, for the first time, that “thick” and “thin” rhombus-shaped tiles could cover a plane, creating a non-repetitive pattern with five-fold rotational symmetry.

Shapes and sizes

Other researchers found that the atoms in certain materials can arrange themselves in similar non-repetitive patterns, which are called quasi-crystals. They are called this because they have a well-defined structure but the atoms are not spaced uniformly as in a normal crystal.

Lu discovered a wealth of girih designs with quasi-crystal patterns through an archive search of documented medieval Islamic architecture. He also found architectural scrolls describing how girih designs were assembled from five regularly shaped tiles, including a bowtie shape, a rhombus, a pentagon, an elongated hexagon, and a decagon.

“These are not quite perfect quasi-crystals,” he told New Scientist, because the patterns show a few defects where a single tile was placed incorrectly. He suspects the defects were mistakes by workers putting together the design specified by the designer. “It’s only 11 defects out of 3700 Penrose tiles, and each can be corrected by a simple rotation,” he says.

The set of five girih tiles decorated with lines that fit together to make regular patterns first appeared about 1200 AD, a time when Islamic mathematics was flowering. The designs grew increasingly complex, and by the 15th century produced near-perfect Penrose patterns found on the Darb-i Imam shrine in Isfahan, Iran.